Compatible system of ℓ-adic representations
Okay kiddo, so let's start with what a representation is. Imagine you have a group of friends, and you want to keep track of how they move around. You can use arrows or pictures to represent their movements. In math, we call this a "representation". It's like a way of showing how the group is acting or moving.
Now, let's talk about "ℓ-adic representations". This is like a special kind of representation that uses numbers to keep track of how the group is moving. The "ℓ" part just means we're using a specific kind of number called an "elliptic curve point". It's like a special type of number that works really well for these kinds of representations.
So, what does it mean for a system of ℓ-adic representations to be "compatible"? Well, imagine you have two groups of friends, and you want to keep track of their movements. You can use two different sets of arrows or pictures to represent each group. But what if some of the friends are in both groups? You want to make sure that their movements in each group match up with each other. That's what it means for the representations to be "compatible" - they have to match up properly for all the friends that are in both groups.
Now, let's put it all together. A "compatible system of ℓ-adic representations" is like a set of representations that all work together nicely. They're like different ways of keeping track of how a group is moving, but they all match up perfectly. It's like if you have a bunch of drawings of your friends, and they all fit together like a puzzle - that's a compatible system of ℓ-adic representations!
Now, let's talk about "ℓ-adic representations". This is like a special kind of representation that uses numbers to keep track of how the group is moving. The "ℓ" part just means we're using a specific kind of number called an "elliptic curve point". It's like a special type of number that works really well for these kinds of representations.
So, what does it mean for a system of ℓ-adic representations to be "compatible"? Well, imagine you have two groups of friends, and you want to keep track of their movements. You can use two different sets of arrows or pictures to represent each group. But what if some of the friends are in both groups? You want to make sure that their movements in each group match up with each other. That's what it means for the representations to be "compatible" - they have to match up properly for all the friends that are in both groups.
Now, let's put it all together. A "compatible system of ℓ-adic representations" is like a set of representations that all work together nicely. They're like different ways of keeping track of how a group is moving, but they all match up perfectly. It's like if you have a bunch of drawings of your friends, and they all fit together like a puzzle - that's a compatible system of ℓ-adic representations!